All of the 5th grade teachers and students from Almond went on a field trip to an art museum. Tickets were $$8.00$ each for teachers and $$2.50$ each for students, and the group paid $$54.50$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$24.00$ each for teachers and $$9.50$ each for students, and the group paid $$181.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+2.5y = 54.5}$ ${24x+9.5y = 181.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-24x-7.5y = -163.5}$ ${24x+9.5y = 181.5}$ Add the top and bottom equations together. $ 2y = 18 $ $ y = \dfrac{18}{2}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {8x+2.5y = 54.5}$ to find $x$ ${8x + 2.5}{(9)}{= 54.5}$ $8x+22.5 = 54.5$ $8x = 32$ $x = \dfrac{32}{8}$ ${x = 4}$ You can also plug ${y = 9}$ into $ {24x+9.5y = 181.5}$ and get the same answer for $x$ ${24x + 9.5}{(9)}{= 181.5}$ ${x = 4}$ There were $4$ teachers and $9$ students on the field trips.